{
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   "source": [
    "# Realization of Recursive Filters\n",
    "\n",
    "*This jupyter notebook is part of a [collection of notebooks](../index.ipynb) on various topics of Digital Signal Processing. Please direct questions and suggestions to [Sascha.Spors@uni-rostock.de](mailto:Sascha.Spors@uni-rostock.de).*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Direct Form Structures\n",
    "\n",
    "The output signal $y[k] = \\mathcal{H} \\{ x[k] \\}$ of a recursive linear-time invariant (LTI) system is given by\n",
    "\n",
    "\\begin{equation}\n",
    "y[k] = \\frac{1}{a_0} \\left( \\sum_{m=0}^{M} b_m \\; x[k-m] - \\sum_{n=1}^{N} a_n \\; y[k-n] \\right)\n",
    "\\end{equation}\n",
    "\n",
    "where $a_n$ and $b_m$ denote constant coefficients and $N$ the order. Note that systems with $M > N$ are in general not stable. The computational realization of above equation requires additions, multiplications, the actual and past samples of the input signal $x[k]$, and the past samples of the output signal $y[k]$. Technically this can be realized by\n",
    "\n",
    "* adders\n",
    "* multipliers, and\n",
    "* unit delays or storage elements.\n",
    "\n",
    "These can be arranged in different topologies. A certain class of structures, which is introduced in the following, is known as *direct form structures*. Other known forms are for instance [cascaded sections](cascaded_structures.ipynb), parallel sections, lattice structures and state-space structures.\n",
    "\n",
    "For the following it is assumed that $a_0 = 1$. This can be achieved for instance by dividing the remaining coefficients by $a_0$."
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Direct Form I\n",
    "\n",
    "The [direct form I](https://en.wikipedia.org/wiki/Digital_filter#Direct_Form_I) is derived by rearranging above equation for $a_0 = 1$\n",
    "\n",
    "\\begin{equation}\n",
    "y[k] = \\sum_{m=0}^{M} b_m \\; x[k-m] + \\sum_{n=1}^{N} - a_n \\; y[k-n]\n",
    "\\end{equation}\n",
    "\n",
    "It is now evident that we can realize the recursive filter by a superposition of a non-recursive and a recursive part. With the elements given above, this results in the following block-diagram\n",
    "\n",
    "![Direct form I filter](direct_form_i.png)\n",
    "\n",
    "This representation is not canonical since $N + M$ unit delays are required to realize a system of order $N$. A benefit of the direct form I is that there is essentially only one summation point which has to be taken care of when considering quantized variables and overflow. The output signal $y[k]$ for the direct form I is computed by realizing above equation.\n",
    "\n",
    "The block diagram of the direct form I can be interpreted as the cascade of two systems. Denoting the signal in between both as $w[k]$ and discarding initial values we get\n",
    "\n",
    "\\begin{align}\n",
    "w[k] &= \\sum_{m=0}^{M} b_m \\; x[k-m] = h_1[k] * x[k] \\\\\n",
    "y[k] &= w[k] + \\sum_{n=1}^{N} - a_n \\; w[k-n] = h_2[k] * w[k] = h_2[k] * h_1[k] * x[k]\n",
    "\\end{align}\n",
    "\n",
    "where $h_1[k] = [b_0, b_1, \\dots, b_M]$ denotes the impulse response of the non-recursive part and $h_2[k] = [1, -a_1, \\dots, -a_N]$ for the recursive part. From the last equality of the second equation and the commutativity of the convolution it becomes clear that the order of the cascade can be exchanged."
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Direct Form II\n",
    "\n",
    "The [direct form II](https://en.wikipedia.org/wiki/Digital_filter#Direct_Form_II) is yielded by exchanging the two systems in above block diagram and noticing that there are two parallel columns of delays which can be combined, since they are redundant. For $N=M$ it is given as\n",
    "\n",
    "![Direct form II filter](direct_form_ii.png)\n",
    "\n",
    "Other cases with $N \\neq M$ can be considered for by setting coefficients to zero. This form is a canonical structure since it only requires $N$ unit delays for a recursive filter of order $N$. The output signal $y[k]$ for the direct form II is computed by the following equations\n",
    "\n",
    "\\begin{align}\n",
    "w[k] &= x[k] + \\sum_{n=1}^{N} - a_n \\; w[k-n] \\\\\n",
    "y[k] &= \\sum_{m=0}^{M} b_m \\; w[k-m]\n",
    "\\end{align}\n",
    "\n",
    "The samples $w[k-m]$ are termed *state* (variables) of a digital filter."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Transposed Direct Form II\n",
    "\n",
    "The block diagrams above can be interpreted as linear [signal flow graphs](https://en.wikipedia.org/wiki/Signal-flow_graph). The theory of these graphs provides useful transformations into different forms which preserve the overall transfer function. Of special interest is the *transposition* or *reversal* of a graph which can be achieved by\n",
    "\n",
    "* exchanging in- and output,\n",
    "* exchanging signal split and summation points, and\n",
    "* reversing the directions of the signal flows.\n",
    "\n",
    "Applying this procedure to the direct form II shown above for $N=M$ yields the transposed direct form II\n",
    "\n",
    "![Transposed direct form II filter](direct_form_ii_t.png)\n",
    "\n",
    "The output signal of the transposed direct form II is given as\n",
    "\n",
    "\\begin{equation}\n",
    "y[k] = b_0 x[k] + \\sum_{m=1}^{M} b_m x[k-n] - \\sum_{n=1}^{N} a_n y[k-n]\n",
    "\\end{equation}\n",
    "\n",
    "Using the signal before the $n$-th delay unit as internal state $w_n[k]$ we can reformulate this into a set of difference equations for computation of the output signal\n",
    "\n",
    "\\begin{align}\n",
    "w_n[k] &= \n",
    "\\begin{cases}\n",
    "w_{n+1}[k-1] - a_n y[k] + b_n x[k] & \\text{for } n=0,1,\\dots,N-1 \\\\\n",
    "-a_N y[k]  + b_N x[k] & \\text{for } n=N\n",
    "\\end{cases}\\\\\n",
    "y[k] &= w_1[k-1] + b_0 x[k]\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Example - Computation of output signal\n",
    "\n",
    "The following example illustrates the computation of the impulse response $h[k]$ of a 2nd-order recursive system using the transposed direct form II as realized by `scipy.signal.lfilter`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
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   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import scipy.signal as sig\n",
    "%matplotlib inline\n",
    "\n",
    "p = 0.90*np.exp(-1j*np.pi/4)\n",
    "a = np.poly([p, np.conj(p)])  # denominator coefficients\n",
    "b = [1, 0, 0]  # numerator coefficients\n",
    "N = 40  # number of computed samples\n",
    "\n",
    "# generate input signal (= Dirac impulse)\n",
    "k = np.arange(N)\n",
    "x = np.where(k == 0, 1.0, 0.0)\n",
    "\n",
    "# filter signal using transposed direct form II\n",
    "h = sig.lfilter(b, a, x)\n",
    "\n",
    "# plot output signal\n",
    "plt.figure(figsize=(8, 4))\n",
    "plt.stem(h, use_line_collection=True)\n",
    "plt.title('Impulse response')\n",
    "plt.xlabel(r'$k$')\n",
    "plt.ylabel(r'$h[k]$')\n",
    "plt.axis([-1, N, -1.5, 1.5])\n",
    "plt.grid()"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise**\n",
    "\n",
    "* Compute the step-response $h_\\epsilon[k] = \\mathcal{H} \\{ \\epsilon[k] \\}$ of the filter by modifying above example.\n",
    "\n",
    "Solution: The step response can be computed by changing the input signal to \n",
    "\n",
    "```python\n",
    "x = np.where(k>=0, 1.0, 0.0)\n",
    "``` \n",
    "\n",
    "or by cumulative summation of the impulse response due to the relation $h_\\epsilon[k] = \\sum_{\\kappa =0}^k h[k]$ of the step response to the impulse response\n",
    "\n",
    "```python\n",
    "he = np.cumsum(h)\n",
    "```"
   ]
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   "source": [
    "**Copyright**\n",
    "\n",
    "This notebook is provided as [Open Educational Resource](https://en.wikipedia.org/wiki/Open_educational_resources). Feel free to use the notebook for your own purposes. The text is licensed under [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/), the code of the IPython examples under the [MIT license](https://opensource.org/licenses/MIT). Please attribute the work as follows: *Sascha Spors, Digital Signal Processing - Lecture notes featuring computational examples*."
   ]
  }
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